STUDY OF FUZZY MEASURE AND SOMEPROPERTIES OF NULL-ADDITIVE
FUZZYMEASUREThesisSubmitted to theKUMAUN UNIVERSITY, NAINITAL,
Uttarakhand, INDIAFOR TE DE!REE OFDoctor of Philosophy in
Mth!"ticsSU"MITTED "YPARUL A#AR$ALM# S$#, M# %hi 'Mathemati$()UNDER
TE SU%ERVISION OFDR% &%S% NAYALA((o$iate %ro*e((orDe+artment o*
mathemati$(!o,t# %o(t-raduate .o&&e-e, Ranikhet
'A&mora)UTTARAKAND#'()*A+,NO$LED#MENTSThe +re$iou( -i*t o*
&earnin- i( a debt that i( di**i$u&t to +a/#
On&/-ratitude$anbe&e*t# Indeed, the0ord( at
m/$ommandareinade1uate in *orm or in (+rit to e2+re(( m/
heart*e&t, dee+ (en(eo* unbound -ratitude and indebtedne(( to
Dr# # S# Na/a&,A((o$iate %ro*e((or-De+artment o*
mathemati$(-!o,t#%o(t-raduate .o&&e-e, Ranikhet
'A&mora)-Uttarakhand#I *ee&-o&den o++ortunit/ and +roud
+ri,i&e-e to 0ork under thein(+irin- -uidan$e, im+erati,e and
*a$ia& (u--e(tion( o* Dr# # S#Na/a&, 0ho i( not 3u(t a
-uide, but an an-e&i$ to me# e ha( noton&/ he&+ed me in
m/ re(ear$h 0ork but a&(o +ro,idedbene,o&ent -uidan$e
durin- the 0ho&e de-ree +ro-ramme# I ha,erea&
a++re$iationandre-ard*orhim*orhi(ma-nanimou(and$oo+erati,eattitude,
+ain(takin-e**ort(, +eer&e(($riti$i(mande(+e$ia&&/ *or
rea&i4in- to $a+ita&i4e on m/ (tren-th to 0ork hardand
a$1uire $on*iden$e to dea& 0ith e,er/ di**i$u&t/#I
amimmen(e&/-rate*u& toDr# Su(hi& 5ainandDr#
%ri/anka%athak, De+artment o* edu$ation-!o,t#
%o(t-raduate.o&&e-e,Ranikhet 'A&mora)-Uttarakhand*or
their ($ho&a(ti$-uidan$e,in(+irin- (u--e(tion( and he&+ at
,ariou( (ta-e( o* the re(ear$h0ork and *or their a&& time
(u++ort, b&e((in-, im+e$$ab&e $oun(e&and $oo+eration
durin- the $our(e o* +re(ent (tud/#I ha,edee+e(t re(+e$t andre-ard(
*or m/*ather, Shri R# N#A-ar0a& and mother, Smt# Ku(um
A-ar0a& *or theiren$oura-ement, moti,ationand,a&uab&e
-uidan$e durin-thi(a$ademi$,enture#
Idon6tha,e0ord(toe2+re((m/-ratitudeand indebtedne(( to m/ brother,
Mr# Saurabh A-ar0a&, 0ho
ha(a&0a/(moti,atedandin(+iredmeto0ork# Fromthedee+e(t$ore o* m/
heart, I e2+re(( ,er/ a**e$tionate thank( to m/ (i(ter,Dr# Nitu
A-ar0a&, 0ho(e emoti,e (u++ort and $are enab&ed me tobrin-
thi( 0ork in (ha+e# The &o,e, a**e$tion and en$oura-ementthat I
-ot *romm/ *ami&/ are un*or-ettab&e# The/ are m/(tren-th,
m/ (u++ort and m/ +ride#I *indm/(e&*
in$a+ab&etoe2+re((m/0arme(t thank(tom/(enior(, 3unior(, and
*riend( *or their en$oura-ement,(u--e(tion(, mora& (u++ort
durin-m/(tud/ +eriod and *or theen3o/ab&e and +re$iou( moment(
(hared 0ith
them#Re(ear$h*e&&o0(hi++ro,idedb/KumaunUni,er(it/,Nainita&durin-+eriodo*
m/do$tora& de-ree+ro-rammei(thank*u&&/a$kno0&ed-ed#
Ranikhet'%aru& A-ar0a&)5une, 789:Authore((PREFA+EIn the
+re(ent the(i( 0e dea& 0ith the (tud/ o* *u44/ mea(ure
andnu&&;additi,e *u44/mea(ure 0hi$hha,e 0ide
a++&i$ation( in$om+uter(, in*ormation te$hno&o-/,
manu*a$turin- (/(tem(,+rodu$tion, $ommuni$ation, tran(+ortation
(/(tem( et$# Fir(ttime, *u44/ (et theor/ introdu$ed b/ L# A# :#
The$entra& $on$e+t
o**u44/mea(uretheor/i(the*u44/mea(ure0hi$h0a(introdu$edb/.ho1uet
in9=:?andinde+endent&/de*ined b/ Su-eno in 9=@A in the $ontent
o* *u44/ inte-ra&(# Theran-e o* nu&&;additi,it/ (et
*un$tion( introdu$ed b/ Ban-#In the *ir(t $ha+ter 0e di($u(( (ome
*undamenta& de*inition( and$on$e+t( re&ated to our re(ear$h
0ork# In thi( $ha+ter 0edi($u((ed about the $on$e+t o* mea(ure,
*u44/, un$ertaint/, *u44/mea(ure andnu&&;additi,it/*u44/
mea(ure# A$$ordin-tothea&ternati,e ,ie0, un$ertaint/ i(
$on(idered e((entia& to ($ien$eC iti(not
on&/anuna,oidab&e+&a-ue, but it ha(, in*a$t,
a-reatuti&it/# Fu44/ mea(ure re+re(ent a t/+e o* un$ertaint/,
0e $ou&da((i-n a ,a&ue to ea$h +o((ib&e $ri(+ (et to
0hi$h the e&ement in1ue(tion mi-ht be&on-# The aim o* the
(e$ond $ha+ter i( to di($u(( the di**eren$e( and(imi&aritie(
bet0een +robabi&it/ mea(ure( and +o((ibi&it/mea(ure(# Thi(
$ha+ter thu( e(tab&i(he( a ne0&ink
bet0een+robabi&it/and+o((ibi&it/theorie(#
Se$ond&/0edi($u(((ome,ariant(o**u44/(ub(et(entro+ie(andthe&inkin-re&ation(o*the(e
1uantitie( 0ithindi$ate a (imi&arit/o*
man/im+ortant$hara$teri(ti$( o* *u44/ (ub(et( entro+ie( and
Shannon6( entro+/#Be ana&/4e ba(i$ +ro+ertie( o* entro+/ o*
*u44/ mea(ure#In the third $ha+ter 0e di($u(( the $onne$tion
bet0een t0o t/+eo* ab(o&ute $ontinuit/ o* a *u44/ mea(ure 0ith
re(+e$t to a -i,ennu&&;additi,e *u44/ mea(ure# A&(o 0e
(tudied (ome theorem( o*Lebe(-uede$om+o(itiont/+e*or
nu&&;additi,e*u44/mea(ure#Be+ro+o(ed-enera&i4ationo*the$on$e+to*(/mmetri$*u44/mea(ure
ba(ed in a de$om+o(ition o* the uni,er(a& (et# At &a(t
0edi($u((ed (ome +ro+ertie( o* *u44/
mea(ure#In*orth$ha+ter0e(ha&&
in,e(ti-ate(tron-re-u&arit/o**u44/mea(ure on metri$ (+a$e(# Be
di($u((ed the re-u&arit/ o* a nu&&;additi,e *u44/
mea(ure and +ro,ed E-oro**6( theoremandLu(in6( theorem*or *u44/
mea(ure( on a metri$ (+a$e# Bedi($u((ed *urther re-u&arit/
+ro+ertie( o* nu&&;additi,e *u44/mea(ure on metri$ (+a$e(#
Be (tudied (ome +ro+ertie( o*innerDouter re-u&arit/ and the
re-u&arit/ o* *u44/ mea(ure undernu&&;additi,it/
$ondition#In the *i*th $ha+ter 0e (tudied (ome theorem( on
the$om+&etene(( o* *u44/mea(ure(+a$eand(ome+ro+ertie( o**u44/
mea(ure and $on,er-en$e in mea(ure# Be di($u(( *inite and
+ure&/ atomi$ *u44/ mea(ure(# In thi( $a(e the
uni*ormauto;$ontinuit/ $an be re+&a$ed 0ith the
nu&&;additi,e# The$on$e+t o* t0o(tru$tura&
$hara$teri(ti$( that are the +(eudo;metri$ -eneratin- +ro+ert/ and
the auto;$ontinuit/ +&a/ im+ortantro&e in the *u44/ mea(ure
theor/#+ont!nts.A%TER 9Intro./ction9#9 Mea(ure9#7 Fu44/9#?
Un$ertaint/9#A Fu44/ mea(ure9#: Nu&&;additi,e *u44/
mea(ure.A%TER 7Possi0ility Th!ory 1!rs/s Pro00ility Th!ory in F/22y
M!s/r! Th!ory7#9 .on$e+t o* a((o$iati,e +robabi&itie( to a
*u44/ mea(ure7#7 Shannon Entro+/ o* *u44/ mea(ure7#? "a(i$
+ro+ertie( o* entro+/ o* *u44/ mea(ure7#A A++&i$ation o*
entro+/ o* *u44/ mea(ure.A%TER ?Prop!rti!s of N/ll-A..iti1! n.
A0sol/t! +ontin/ity ofF/22y M!s/r!?#9 Ab(o&ute $ontinuit/ o* a
*u44/ mea(ure?#7 Some re(u&t( on nu&&;additi,e *u44/
mea(ure?#? Lebe(-ue de$om+o(ition t/+e nu&&;additi,e *u44/
mea(ure?#A !enera&i4ation o* the (/mmetri$ *u44/ mea(ure?#:
%ro+ertie( o* *u44/ mea(ure(.A%TER AProp!rti!s of Stron3 R!3/lrity
of F/22y M!s/r! on M!tric Spc!A#9 Nu&& additi,e *u44/
mea(ure on metri$ (+a$eA#7 Stron- re-u&arit/ o* *u44/ mea(ure
on metri$ (+a$eA#? %ro+ertie( o* innerEouter re-u&arit/ o*
*u44/ mea(ure.A%TER :+ontin/o/s- /to-contin/o/s n. co"pl!t!n!ss of
f/22y"!s/r!:#9 .ontinuou( and auto;$ontinuou( o* *u44/ mea(ure:#7
Re(u&t( on $om+&etene(( o* *u44/ mea(ure (+a$e:#?
.on,er-en$e in *u44/ mea(ure:#A Some other +ro+ertie( o* *u44/
mea(ure and $on,er-en$e"I"LIO!RA%Y%ub&i(hed Re(ear$h %a+er(
